What are Co-Prime Numbers? Definition, Examples, and Properties (1 to 100)

Co Prime Numbers

Building on our previous guide to Prime numbers—which are integers with exactly two factors: 1 and themselves—we will now explore the concept of Co-Prime Numbers. Simply put, co-prime numbers are two numbers that share only 1 as a common factor. Understanding this concept begins with identifying common factors, which are whole numbers that divide two or more integers exactly. For instance, the only common factor between 2 and 3 is 1. Here is why:

2 = 1 × 2 and 3 = 1 × 3. Since 1 is the only shared factor, these numbers are co-prime. 

Identifying co-prime numbers is straightforward when you evaluate their common factors. If the greatest common divisor (GCD) or highest common factor (HCF) of two numbers is exactly 1, the pair is considered co-prime. In this article, we will walk you through clear examples, essential properties, and simple methods to determine if any given pair of numbers is co-prime.

Co Prime Number Definition

Co-prime numbers are often referred to as Relatively Prime Numbers. It is a common misconception that numbers must be prime to be co-prime; however, this is not true. Composite numbers can also be co-prime as long as their only shared factor is 1.

Examples:

Example 1: 8 and 15

Factors of 8: 1, 2, 4, 8

Factors of 15: 1, 3, 5, 15 

Because their only common factor is 1, 8 and 15 are co-prime numbers.

Example 2: 3 and 5

Factors of 3: 1, 3

Factors of 5: 1, 5

3 and 5 are co-prime as 1 is their only shared factor.

Example 3: 4 and 6

Factors of 4: 1, 2, 4

Factors of 6: 1, 2, 3, 6

Since they share 1 and 2 as common factors, 4 and 6 are not co-prime.

Co Prime Number Examples 

Co-prime numbers always function in pairs. Here are additional examples of co-prime sets:

(2, 9) Factors of 2: 1, 2 Factors of 9: 1, 3, 9	(3, 7) Factors of 3: 1, 3 Factors of 7: 1, 7	(2, 47) Factors of 2: 1, 2 Factors of 47: 1, 47
(1, 42) Factor of 1: 1 Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42	(11, 13) Factors of 11: 1, 11 Factors of 13: 1, 13	(14, 25) Factors of 14: 1, 2, 7, 14 Factors of 25: 1, 5, 25
(15, 32) Factors of 15: 1, 3, 5, 15 Factors of 32: 1, 2, 4, 8, 16, 32	(1, 10) Factor of 1: 1 Factors of 10: 1, 2, 5, 10	(5, 21) Factors of 5: 1, 5 Factors of 21: 1, 3, 7, 21

Since all these pairs have only 1 as a shared factor, they are all co-prime.

How to Identify Co-prime Numbers?

To identify whether a pair of numbers is co-prime, follow these simple steps:

 Step 1: List all the factors for each number.

 Step 2: Identify all common factors.

 Step 3: Check if the highest common factor is 1. If it is, the numbers are co-prime.

For example, let's test 6 and 9:

Step 1: Factors of 6 and 9 are:

6 = 1, 2, 3, 6

9 = 1, 3, 9

Step 2: Common factors are 1 and 3.

Step 3: Since the HCF is 3, the pair is not co-prime.

Properties of Co-prime Numbers

Properties of co-prime numbers include:

• A co-prime number pair can be formed by (1) with any number.

Numbers like (1, 2), (1, 3), (1, 8), (1, 15) are always co-prime.

• Two even numbers can never be a co-prime number because they always have two common factors i.e. 1 and 2.

Pairs like (2, 4), (8, 10), (20, 22) are not, as they share factors other than 1.

• The sum of two co-prime numbers is always coprime with the product of those numbers.

Example: For (3, 8), the sum is 11 and the product is 24; (11, 24) are also co-prime.

• The Highest Common Factor (HCF) of two co-prime numbers is always 1.

Example: (9, 11) are co-prime with an HCF of 1.

• The Least Common Multiple (LCM) of two co-prime numbers is always their product.

Example: (9, 11) have an LCM of 9 × 11 = 99.

• All pairs of the two consecutive numbers are always coprime numbers and their common factor is always 1.

Examples of consecutive integers that are co-prime: (2,3), (5,6), (10,11), (15,16), (16,17), (58,59), (100,101).

• Two Prime numbers are always coprime numbers because they have only one common factor i.e. 1.

Example: (11, 13) are co-prime.

Example: (17, 29) are co-prime.

Example: (31, 37) are co-prime.

All these pairs have an HCF of 1, confirming they are co-prime.

Difference between Co Prime and Twin Prime Numbers

Distinguishing from 'Twin Primes': While co-primes have an HCF of 1, twin primes are a specific subset of primes with a difference of 2.

For example, 3 and 5 are prime numbers that are also twin primes (5 - 3 = 2), as are 5 and 7.

Difference between Co-prime and Twin Prime Numbers
Co-prime Number	Twin Prime Numbers
The difference of two co-primes can be any whole number.	The difference between two twin primes is always 2.
Co-prime numbers can be prime or composite numbers.	Twin prime numbers are always prime numbers.
Co-prime numbers may or may not be twin prime numbers.	All the sets of twin prime numbers are co-prime.
1 can form a coprime pair with any number.	1 forms a twin prime pair only with 3. Ie 1 and 3 as (3-1=2)

Important Points on Co-prime Numbers:

1. The co-prime number can not necessarily be a prime number. It can be a composite number also.
2. Any two given prime numbers must be a Co-prime.
3. Sets of two even numbers will never be a co-prime number.
4. Number 1 (one) can make a coprime pair with any number.
5. Any two consecutive numbers are always a co-prime number.

Co Prime Numbers Questions

Q1: Which of the following pairs are co-prime? (8, 9), (11, 12), (21, 9)

Answer: We find the HCF for each.

For 8 and 9:

Factors of 8: 1, 2, 4, 8

Factors of 9: 1, 3, 9

HCF is 1, so they are co-prime.

For 11 and 12:

Factors of 11: 1, 11

Factors of 12: 1, 2, 3, 4, 6, 12

HCF is 1, so they are co-prime.

For 21 and 9:

Factors of 21: 1, 3, 7, 21

Factors of 9: 1, 3, 9

Since they share 1 and 3, they are not co-prime.

Q2: What is the difference between Prime and Co-prime numbers?

Answer: Prime numbers are individual numbers with exactly two factors (1 and itself).

Co-prime numbers are pairs that share only 1 as a common factor.

Q3: Is there a number that is co-prime to all others?

Answer: Yes, 1 is co-prime to every integer because 1 is the only factor it shares with any other number.

Q4: Are 250 and 148 co-prime?

Answer: Let's look at their factors:

250 factors include 1, 2, 5, 10, 25, 50, 125, 250.

148 factors include 1, 2, 4, 37, 74, 148.

Since they share 1 and 2, they are not co-prime.

General tip: Any pair of even numbers will never be co-prime, as they always share 2 as a factor.

Q5: Are 40 and 78 co-prime?

Answer: Since both 40 and 78 are even numbers, they share 2 as a factor and are therefore not co-prime.

Q6: If a and b are co-prime, are their cubes (a³ and b³) also co-prime?

Answer: Yes, they are.

Take a=5 and b=7:

5³ = 125

7³ = 343

125 and 343 share no factors other than 1, so they are co-prime.

Factorization:

125 = 1, 5, 25, 125

343 = 1, 7, 49, 343

The HCF remains 1.

Q7: From 23, 25, 12, and 18, identify all co-prime pairs.

Answer: Two numbers are co-prime if their HCF is 1.

Factors:

23 = 1, 23

25 = 1, 5, 25

12 = 1, 2, 3, 4, 6, 12

18 = 1, 2, 3, 6, 9, 18

Possible co-prime sets:

(23, 25), (23, 12), (23, 18), (25, 12), (25, 18)

Q8: Find the HCF of the following sets.

1. 32 and 43
2. 28 and 64

Answer: Listing factors:

1. 32 and 43

32 = 1, 2, 4, 8, 16, 32

43 = 1, 43

HCF = 1

1. 28 and 64

28 = 1, 2, 4, 7, 14, 28

64 = 1, 2, 4, 8, 16, 32, 64

Common factors are 1, 2, 4.

HCF = 4.

Q9: Find the largest number that divides both 225 and 340.

Answer: We need the HCF.

Factoring 225 and 340:

225 = 1, 3, 5, 9, 15, 25, 45, 75, 225

340 = 1, 2, 4, 5, 10, 17, 20, 34, 68, 85, 170, 340

Common factors are 1 and 5.

The largest number is 5.

Q10: Show that 61 and 67 are co-prime.

Answer: They must share only 1 as a factor.

Factorization:

61 = 1, 61

67 = 1, 67

Since the HCF is 1, they are co-prime.

Any pair of prime numbers is inherently co-prime.